Bits or symbols are transmitted in the form of pulses in digital communication systems. Pulse shaping filters are used in order to limit the bandwidth required for the transmission of the pulses. One typical representative of this class of filters is the so-called raised cosine pulse shaping filter. The name of the filter is derived from the profile of the transfer function. If separate transmission and reception filters are used rather than a transmission-end raised cosine pulse shaping filter, these are based on so-called root raised cosine filters. In this case, the square of the transfer function of a root raise cosine filter corresponds to the transfer function of a raised cosine filter. More detailed information relating to pulse shaping filters can be found in the article “The care and feeding of digital, pulse-shaping-filters” by Ken Gentile, RF Design, April 2004.
For modern radio systems with high data transmission rates, the use of pulse shaping filters is advisable in order to reduce the required transmission bandwidth, owing to the limited overall transmission capacity of the transmission medium. High data transmission rates are made possible, for example, by the use of 4-value or 8-value modulation methods. The enhanced Bluetooth 1.2 Standard with EDR (enhanced data rate) uses π/4-DQPSK or 8 DPSK modulation (differential (quadrature) phase shift keying) for increased data transmission rates of 2 Mb/s and 3 Mb/s, with π/4-DQPSK representing a 4-value type of modulation, and 8 DPSK modulation representing an 8-value type of modulation.
With modulation methods such as these, 1d(M) bits of the datastream to be transmitted are mapped onto a complex symbol, depending on the significance M of the modulation method, and this is also referred to as symbol mapping. A complex symbol such as this in this case forms a point on the unit circle with a real part and an imaginary part. In this case, the real part corresponds to the in-phase component (I component for short), and the imaginary part corresponds to the quadrature component (Q component for short). The subsequent pulse shaping, in particular by means of two root raised cosine filters, is carried out generally separately in two paths for the I component and the Q component. In general, an increase in the sampling rate, with, for example, double, quadruple or octuple oversampling is carried out before the actual pulse shaping filter. Such a sampling rate increase can be achieved by repetition of the sample value. The need for a sampling rate increase is justified because the Nyquist condition means that the cut-off frequency of the transfer function is limited to half the sampling rate. If the cut-off frequency of the transfer function is thus, for example, the same as the symbol rate fs, at least double oversampling is required.
Pulse shaping filters are generally in the form of digital FIR filters. The expression FIR (finite impulse response) indicates that the impulse response has a finite length. FIR filters known from the prior art are based on digital adders, digital multipliers and delay elements. FIGS. 1a and 1b show two alternative embodiments of FIR filters. According to the first embodiment shown in FIG. 1a, an input signal X is delayed by a chain of N′−1 delay elements. The chain comprises N′ signal taps, with the signal being multiplied at each signal tap by a coefficient αi—where i=1, 2, . . . , N′—of the impulse response by means of a multiplier. The multiplied signals are added in N′−1 adders. Alternatively, the addition operations and the delay operations can be interchanged, thus resulting in the implementation of an FIR filter as illustrated in FIG. 1b. The fact that two different embodiments are feasible is justified by the fact that the convolution is a commutative operation.